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6.11. Exercises 61
x=1
y=x+1
print(c(x, y+3, x+y))
Exercise 6.2. The Ackermann function, A(m, n), is defined:
n + 1 if m = 0 A(m,n)= A(m−1,1) ifm>0andn=0 A(m−1,A(m,n−1)) ifm>0andn>0.
See http: // en. wikipedia. org/ wiki/ Ackermann_ function . Write a function named ack thatevaluatestheAckermannfunction.Useyourfunctiontoevaluateack(3, 4),whichshouldbe 125. What happens for larger values of m and n? Solution: http: // thinkpython2. com/ code/ ackermann. py .
Exercise 6.3. A palindrome is a word that is spelled the same backward and forward, like “noon” and “redivider”. Recursively, a word is a palindrome if the first and last letters are the same and the middle is a palindrome.
The following are functions that take a string argument and return the first, last, and middle letters:
def first(word):
return word[0]
def last(word):
return word[-1]
def middle(word):
return word[1:-1]
We’ll see how they work in Chapter 8.
1. Type these functions into a file named palindrome.py and test them out. What happens if you call middle with a string with two letters? One letter? What about the empty string, which is written '' and contains no letters?
2. Write a function called is_palindrome that takes a string argument and returns True if it is a palindrome and False otherwise. Remember that you can use the built-in function len to check the length of a string.
Solution: http: // thinkpython2. com/ code/ palindrome_ soln. py .
Exercise 6.4. A number, a, is a power of b if it is divisible by b and a/b is a power of b. Write a function called is_power that takes parameters a and b and returns True if a is a power of b. Note: you will have to think about the base case.
Exercise 6.5. The greatest common divisor (GCD) of a and b is the largest number that divides both of them with no remainder.
One way to find the GCD of two numbers is based on the observation that if r is the remainder when a is divided by b, then gcd(a, b) = gcd(b, r). As a base case, we can use gcd(a, 0) = a.
Write a function called gcd that takes parameters a and b and returns their greatest common divisor.
Credit: This exercise is based on an example from Abelson and Sussman’s Structure and Interpre- tation of Computer Programs.